Vacuum Instability in Strong Fields

An electric background can turn an incoming matter vacuum into a state containing particle–antiparticle pairs. The probability that no pairs are present at the end is the vacuum persistence probability. Its negative logarithm is related to the imaginary part of an in/out effective action. It is generally different from the mean number of produced pairs. This page derives that distinction and explains the constant-electric-field result within the prescribed-background, one-loop approximation.

Required background. Pair Creation defines in/out modes and their occupations; Antiparticles fixes charge counting.

Helpful background. Klein Paradox supplies the electric-barrier picture; Pair-Creation Thresholds separates thresholds from field-strength scales.

Vacuum survival in a single pair channel

First take a finite set of independent paired modes of a quantized matter field in a classical source. Start in the incoming vacuum. In the paired basis of the Bogoliubov construction, let $N=|\beta|^2$ be the outgoing mean number of pairs in one channel.

A fermionic channel can contain either zero or one pair. Its probabilities are therefore

$$P_0=1-N,\qquad P_1=N.$$

A bosonic channel can contain any number $n\geq0$. Write the incoming vacuum in the outgoing number basis as

$$|0,{\rm in}\rangle_\lambda =\sum_{n=0}^{\infty}c_n|n_a=n,n_b=n;{\rm out}\rangle.$$

The inverse operator transformation gives $a_{\rm in}=\alpha^*a_{\rm out}-\beta b_{\rm out}^\dagger$. Annihilating the incoming vacuum implies $c_{n+1}=(\beta/\alpha^*)c_n$. Since $|\alpha|^2=1+N$, normalization yields

$$P_n=\frac{N^n}{(1+N)^{n+1}}, \qquad P_0=\frac{1}{1+N}.$$

This geometric distribution has mean $N$ and variance $N(1+N)$. The fermionic variance is $N(1-N)$. The distinction arises from statistics even though the vacuum mean is called $|\beta|^2$ in both cases.

Many channels and the imaginary effective action

For independent channels labeled by $\lambda$, the probability of no outgoing pairs is

$$\begin{aligned} P_{\rm vac}^{\rm F}&=\prod_\lambda(1-N_\lambda),\\ P_{\rm vac}^{\rm B}&=\prod_\lambda(1+N_\lambda)^{-1}. \end{aligned}$$

Independence here follows from the quadratic matter evolution in the prescribed background after choosing the paired mode basis. It should not be assumed for a general interacting many-particle state. The mean pair count is instead

$$\overline n_{\rm pairs}=\sum_\lambda N_\lambda.$$

Define the normalized vacuum amplitude by $\langle0,{\rm out}|0,{\rm in}\rangle=e^{i\Gamma/\hbar}$. Its phase can depend on vacuum conventions, but its modulus gives

$$P_{\rm vac}=e^{-2\,\operatorname{Im}\Gamma/\hbar}, \qquad -\ln P_{\rm vac}=\frac{2\,\operatorname{Im}\Gamma}{\hbar}.$$

The mode products consequently imply

$$\begin{aligned} -\ln P_{\rm vac}^{\rm F} &=\sum_\lambda\left(N_\lambda+\frac{N_\lambda^2}{2} +\frac{N_\lambda^3}{3}+\cdots\right),\\ -\ln P_{\rm vac}^{\rm B} &=\sum_\lambda\left(N_\lambda-\frac{N_\lambda^2}{2} +\frac{N_\lambda^3}{3}-\cdots\right). \end{aligned}$$

These series require the relevant small-occupation convergence conditions; the logarithmic expressions remain the starting point. In a dilute collection of channels, $-\ln P_{\rm vac}\simeq\overline n_{\rm pairs}$. Beyond that approximation they differ. In particular, $1-P_{\rm vac}$ is the probability of at least one pair, and is a third quantity.

For example, two fermionic channels with occupations $N_1=1/4$, $N_2=1/2$ have $\overline n_{\rm pairs}=3/4$, $P_{\rm vac}=3/8$, and probability $5/8$ for at least one pair. Neither $3/4$ nor $5/8$ equals $-\ln(3/8)$.

A constant electric field and its tunneling exponent

Consider a uniform electric field of magnitude $\mathcal E>0$ in $3+1$ dimensions, acting for a long time $T$ over volume $\mathcal V$. Use switching and finite-volume regulators before taking the bulk limit. The matter has mass $m>0$ and charge magnitude $|q|$. Ignore backreaction and additional radiative corrections.

The exponent has a simple semiclassical interpretation. In a static electric gauge, the local kinetic energy $\epsilon(x)=E-q\Phi(x)$ varies linearly with $|d\epsilon/dx|=|q|\mathcal E$. At zero transverse momentum, the classically forbidden interval is $-mc^2<\epsilon<mc^2$, with imaginary longitudinal momentum

$$|p_x|=\frac{1}{c}\sqrt{m^2c^4-\epsilon^2}.$$

The barrier exponent for a probability is twice the amplitude action:

$$\begin{aligned} \frac{2}{\hbar}\int |p_x|\,dx &=\frac{2}{\hbar c|q|\mathcal E} \int_{-mc^2}^{mc^2} \sqrt{m^2c^4-\epsilon^2}\,d\epsilon\\ &=\frac{\pi m^2c^3}{|q|\hbar\mathcal E}. \end{aligned}$$

Thus production contains the suppression $\exp(-\pi\mathcal E_{\rm crit}/\mathcal E)$, where

$$\mathcal E_{\rm crit}=\frac{m^2c^3}{|q|\hbar}.$$

This estimates the exponential factor, not its degeneracy, prefactor, or multipair statistics. Those need the normalized field calculation. In particular, $\mathcal E_{\rm crit}$ is a characteristic scale, not an exact onset below which production vanishes.

Mean pairs and the Schwinger series

Return to $\hbar=c=1$. In the long-duration uniform-field limit the normalized single-channel result is

$$N_{\mathbf p_\perp} =\exp\left[-\frac{\pi(m^2+p_\perp^2)}{|q|\mathcal E}\right].$$

The longitudinal canonical momentum interval swept during time $T$ has width $|q|\mathcal E T$. With spatial mode density $\mathcal V/(2\pi)^3$, this gives

$$\begin{aligned} \frac{\overline n_{\rm pairs}}{\mathcal V T} &=\frac{g_s|q|\mathcal E}{(2\pi)^3} \int d^2p_\perp\,N_{\mathbf p_\perp}\\ &=\frac{g_s(|q|\mathcal E)^2}{8\pi^3} e^{-\pi m^2/(|q|\mathcal E)}. \end{aligned}$$

Here $g_s=2$ for a Dirac field and $g_s=1$ for a complex scalar. This counts pairs, with each spin channel counted once, rather than the sum of particle and antiparticle counts.

Define the vacuum-decay exponent per spacetime volume by $w=-\ln P_{\rm vac}/(\mathcal V T)$. Insert the same occupations into the logarithms above. At order $n$ in their series the transverse Gaussian integral is

$$\int d^2p_\perp\, e^{-n\pi p_\perp^2/(|q|\mathcal E)} =\frac{|q|\mathcal E}{n}.$$

Together with the logarithm’s $1/n$, this gives the constant-field Schwinger expressions

$$w_{\rm F}= \frac{(|q|\mathcal E)^2}{4\pi^3} \sum_{n=1}^{\infty} \frac{e^{-n\pi m^2/(|q|\mathcal E)}}{n^2},$$ $$w_{\rm B}= \frac{(|q|\mathcal E)^2}{8\pi^3} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}e^{-n\pi m^2/(|q|\mathcal E)}}{n^2}.$$

They equal $2\operatorname{Im}\mathcal L_{\rm eff}$ in natural units in the homogeneous limit. The integer $n$ in these logarithmic series does not label an exclusive probability for exactly $n$ pairs. Such probabilities come from the mode distributions and their convolution.

For a Dirac field, the SI prefactor in $w_{\rm F}$ is $(|q|\mathcal E)^2/(4\pi^3\hbar^2c)$ and its exponent is $n\pi\mathcal E_{\rm crit}/\mathcal E$. Its units are inverse volume per time. Keeping only $n=1$ reproduces the mean-pair rate written above with units restored. The full $w_{\rm F}$ exceeds that mean; the full bosonic $w_{\rm B}$ is smaller than its mean. Both statements follow already from $-\ln(1-N)>N$ and $\ln(1+N)<N$ for $N>0$.

What the background approximation leaves out

The constant-field result is a bulk limit, not an arbitrary pulse formula. Finite duration, finite spatial extent, switching, frequency content, and gradients can change the spectrum and production mechanism. In a static finite electric region, its available potential drop matters as well as the local field strength. For a long pulse, accumulated matter current and energy loss can invalidate a prescribed undepleted source even if a one-loop calculation of its initial production is accurate.

A static uniform magnetic field by itself does not have this electric vacuum instability for minimally coupled massive scalar or Dirac matter in flat spacetime. It changes the spectrum into Landau levels without supplying electric work. A single ideal plane electromagnetic wave in vacuum likewise does not produce massive pairs from vacuum by itself; another wave, particle, or background changes that kinematic setting.

Finally, $P_{\rm vac}$ can tend to zero as $\mathcal V T$ grows while $w$ stays finite. That extensive limit does not mean that the vacuum disappears instantaneously at every point. Nor does a finite-volume Bogoliubov transformation guarantee a global unitary map between infinite-volume Fock representations.

Exercises

Exclusive probabilities. For the two fermionic channels $N_1=1/4$, $N_2=1/2$, calculate the probabilities for exactly zero, one, and two pairs.

Solution

They are $P_0=(3/4)(1/2)=3/8$, $P_1=(1/4)(1/2)+(3/4)(1/2)=1/2$, and $P_2=(1/4)(1/2)=1/8$. Their sum is one and $P_1+2P_2=3/4$ is the mean pair count.

A large bosonic mean. A channel has $N=2$. Find its vacuum probability, probability for exactly one pair, and probability for at least one pair.

Solution

$P_0=1/3$, $P_1=2/9$, and $1-P_0=2/3$. The mean two is compatible with all probabilities lying between zero and one because occupations are unbounded.

First correction to the dilute limit. Put $z=e^{-\pi m^2/(|q|\mathcal E)}\ll1$. Find the ratio of $w$ to the mean-pair rate for each statistic through first order in $z$.

Solution

After canceling the respective prefactors, $w_{\rm F}/r_{\rm F}=1+z/4+O(z^2)$ and $w_{\rm B}/r_{\rm B}=1-z/4+O(z^2)$. The $1/4$ includes both the logarithm and transverse momentum integration; it is not the single-channel coefficient $1/2$.

References

• Dunne, Gerald V. “Heisenberg–Euler Effective Lagrangians: Basics and Extensions.” In From Fields to Strings: Circumnavigating Theoretical Physics, edited by M. Shifman, A. Vainshtein, and J. Wheater. World Scientific (2005). doi:10.1142/9789812775344_0014; 2004 manuscript. Constant-field spinor/scalar results and the limits of extensions to nonuniform backgrounds.
• Gavrilov, S. P., and D. M. Gitman. “Vacuum Instability in External Fields.” Physical Review D 53, 7162–7175 (1996). doi:10.1103/PhysRevD.53.7162; author manuscript. In/out occupation numbers, vacuum persistence, and regulated background-field limits.
• Schwinger, Julian. “On Gauge Invariance and Vacuum Polarization.” Physical Review 82, 664–679 (1951). doi:10.1103/PhysRev.82.664. Effective-action treatment of the constant electromagnetic background.